
From the point of view of low-dimensional topology, we consider the mapping torus \(M_{\varphi }\)  of \(\varphi \) , which is a three-manifold with \(b_1=1\) ;

On the algebraic geometry side, we consider a theta characteristic of \(\Sigma \) , i.e. one of the \(2^{2g}\)  holomorphic line bundles \(L\)  which are square roots of the canonical bundle \(K\) , i.e. \(L^2\cong K\)  [1]}.

